When I was a kid, I absolutely hated chicken killing day but was often the assigned without-clout person to hold the chicken by the feet while my mother inserted the neck between the two nails protruding from the chop block and wielded a swift hatchet and then to fling the body off a distance.

Much as we hated the job (the brother just older than I would hide under a building until the job was done), today I think that cutting off the head was the least traumatic death for the chicken.

And of course Deckman is correct - the body has to be bled- cutting off the head served both functions.